By Theorem \(\ref {gammacongruence}\) ,
\(\Phi _{E_1}\cdot \chi _t(\Gamma , E_1)=1\Leftrightarrow \Phi _{E_2}\cdot \chi _t(\Gamma ,E_2)=1.\) 
By Theorem \(\ref {GL2toGamma}\) ,
\(\chi _t(G_i, E_i)\sim \chi _t(\Gamma , E_i)\times \prod _{l\in \mathcal {M}_{E_i}} L_l(E_i,1)^{-1}.\) 
By [1]}, a prime \(l\in \mathcal {M}_{E_i}\)  if and only if \(E_i\)  does not have potentially good reduction at \(l\) . By our assumption, at every prime \(l\in \mathcal {M}_{E_i}\) , we have that \(L_l(E_i,1)\)  is a \(p\) -adic unit. The assertion follows from this.
